Realizability and recursive mathematics

Section 1: Philosophy, logic and constructivityPhilosophy, formal logic and the theory of computation all bear on problems in the foundations of constructive mathematics. There are few places where these, often competing, disciplines converge more neatly than in the theory of realizability structures. Uealizability applies recursion-theoretic concepts to give interpretations of constructivism along lines suggested originally by Heyting and Kleene. The research reported in the dissertation revives the original insights of Kleene—by which realizability structures are viewed as models rather than proof-theoretic interpretations—to solve a major problem of classification and to draw mathematical consequences from its solution.Section 2: Intuitionism and recursion: the problem of classificationThe internal structure of constructivism presents an interesting problem. Mathematically, it is a problem of classification; for philosophy, it is one of conceptual organization. Within the past seventy years, constructive mathematics has grown into a jungle of fullydeveloped "constructivities," approaches to the mathematics of the calculable which range from strict finitism through hyperarithmetic model theory. The problem we address is taxonomic: to sort through the jungle, set standards for classification and determine those features which run through everything that is properly "constructive."There are two notable approaches to constructivity; these must appear prominently in any proposed classification. The most famous is Brouwer's intuitioniam. Intuitionism relies on a complete constructivization of the basic mathematical objects and logical operations. The other is classical recursive mathematics, as represented by the work of Dekker, Myhill, and Nerode. Classical constructivists use standard logic in a mathematical universe restricted to coded objects and recursive operations.The theorems of the dissertation give a precise answer to the classification problem for intuitionism and classical constructivism. Between these realms arc connected semantically through a model of intuitionistic set theory. The intuitionistic set theory IZF encompasses all of the intuitionistic mathematics that does not involve choice sequences. (This includes all the work of the Bishop school.) IZF has as a model a recursion-theoretic structure, V(A7), based on Kleene realizability. Since realizability takes set variables to range over "effective" objects, large parts of classical constructivism appear over the model as inter¬ preted subsystems of intuitionistic set theory. For example, the entire first-order classical theory of recursive cardinals and ordinals comes out as an intuitionistic theory of cardinals and ordinals under realizability. In brief, we prove that a satisfactory partial solution to the classification problem exists; theories in classical recursive constructivism are identical, under a natural interpretation, to intuitionistic theories. The interpretation is especially satisfactory because it is not a Godel-style translation; the interpretation can be developed so that it leaves the classical logical forms unchanged.Section 3: Mathematical applications of the translation:The solution to the classification problem is a bridge capable of carrying two-way mathematical traffic. In one direction, an identification of classical constructivism with intuitionism yields a certain elimination of recursion theory from the standard mathematical theory of effective structures, leaving pure set theory and a bit of model theory. Not only are the theorems of classical effective mathematics faithfully represented in intuitionistic set theory, but also the arguments that provide proofs of those theorems. Via realizability, one can find set-theoretic proofs of many effective results, and the set-theoretic proofs are often more straightforward than their recursion-theoretic counterparts. The new proofs are also more transparent, because they involve, rather than recursion theory plus set theory, at most the set-theoretic "axioms" of effective mathematics.Working the other way, many of the negative ("cannot be obtained recursively") results of classical constructivism carry over immediately into strong independence results from intuitionism. The theorems of Kalantari and Retzlaff on effective topology, for instance, turn into independence proofs concerning the structure of the usual topology on the intuitionistic reals.The realizability methods that shed so much light over recursive set theory can be applied to "recursive theories" generally. We devote a chapter to verifying that the realizability techniques can be used to good effect in the semantical foundations of computer science. The classical theory of effectively given computational domains a la Scott can be subsumed into the Kleene realizability universe as a species of countable noneffective domains. In this way, the theory of effective domains becomes a chapter (under interpre¬ tation) in an intuitionistic study of denotational semantics. We then show how the "extra information" captured in the logical signs under realizability can be used to give proofs of classical theorems about effective domains.Section 4: Solutions to metamathematical problems:The realizability model for set theory is very tractible; in many ways, it resembles a Boolean-valued universe. The tractibility is apparent in the solutions it offers to a number of open problems in the metamathematics of constructivity. First, there is the perennial problem of finding and delimiting in the wide constructive universe those features that correspond to structures familiar from classical mathematics. In the realizability model, it is easy to locate the collection of classical ordinals and to show that they form, intuitionistically, a set rather than a proper class. Also, one interprets an argument of Dekker and Myhill to prove that the classical powerset of the natural numbers contains at least continuum-many distinct cardinals.Second, a major tenet of Bishop's program for constructivity has been that constructive mathematics is "numerical:" all the properties of constructive objects, including the real numbers, can be represented as properties of the natural numbers. The realizability model shows that Bishop's numericalization of mathematics can, in principle, be accomplished. Every set over the model with decidable equality and every metric space is enumerated by a collection of natural numbers

The continuum hypothesis : independence and truth-value

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Philosophy, 1974.MIT Humanities Library copy: issued in two vols.Leaf number 84 used twice. Also issued as a two-volume set.Includes bibliographical references (leaves 217-258).by Thomas S. Weston.Ph.D

Hilbert's thesis : some considerations about formalizations of mathematics

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Linguistics and Philosophy, 1982.MICROFICHE COPY AVAILABLE IN ARCHIVES AND HUMANITIESBibliography: leaves 175-176.by Lon A. Berk.Ph.D

Second Order Logic and Logical Form

This thesis explores several related issues surrounding second order logic. The central problem running throughout is whether second order logic should provide the underlying logic for formalizations of natural language. A prior problem is determining the significance of this choice. Such controversies over the adoption of a logic usually involve assessing the merits of challengers to first order logic. In some of these rival systems various first order logical truths do not hold. The failure of the Law of the Excluded Middle in intuitionistic systems is the most common example. The other alternatives to first order logic accept it as a part of the truth, but extend it by adding new logical constants. Some modal systems of logic are formed by adding to first order logic a symbol intended to be read as \u27it is logically necessary that.\u27 The first order semantics is extended to provide truth conditions for sentences containing this new symbol. In such cases the debate is whether we are justified in expanding the list of logical constants provided by first order logic. We accept the first order logical constants and are deciding whether, e.g., \u27it is logically necessary that\u27 should be added to the list

The crucial role of proof--a classical defense against mathematical empiricism

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Linguistics and Philosophy, 1993.Includes bibliographical references (leaves 135-137).by Catherine Allen Womack.Ph.D

From axiomatization to generalizatrion of set theory

The thesis examines the philosophical and foundational significance of Cohen's Independence results. A distinction is made between the mathematical and logical analyses of the "set" concept. It is argued that topos theory is the natural generalization of the mathematical theory of sets and is the appropriate foundational response to the problems raised by Cohen's results. The thesis is divided into three parts. The first is a discussion of the relationship between "informal" mathematical theories and their formal axiomatic realizations this relationship being singularly problematic in the case of set theory. The second part deals with the development of the set concept within the mathemtical approach. In particular Skolem's reformulation of Zermlelo's notion of "definite properties". In the third part an account is given of the emergence and development of topos theory. Then the considerations of the first two parts are applied to demonstrate that the shift to topos theory, specifically in its guise of LST (local set theory), is the appropriate next step in the evolution of the concept of set, within the mathematical approach, in the light of the significance of Cohen's Independence results

Logic and the Challenge of Computer Science

https://deepblue.lib.umich.edu/bitstream/2027.42/154161/1/39015099114889.pd